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On the Exact Matching Problem in Dense Graphs

Authors Nicolas El Maalouly , Sebastian Haslebacher , Lasse Wulf

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Author Details

Nicolas El Maalouly
  • Department of Computer Science, ETH Zurich, Switzerland
Sebastian Haslebacher
  • Department of Computer Science, ETH Zurich, Switzerland
Lasse Wulf
  • Institute of Discrete Mathematics, TU Graz, Austria

Funding

  • Wulf, Lasse: Supported by the Austrian Science Fund (FWF): W1230.

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Nicolas El Maalouly, Sebastian Haslebacher, and Lasse Wulf. On the Exact Matching Problem in Dense Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.33

Abstract

In the Exact Matching problem, we are given a graph whose edges are colored red or blue and the task is to decide for a given integer k, if there is a perfect matching with exactly k red edges. Since 1987 it is known that the Exact Matching Problem can be solved in randomized polynomial time. Despite numerous efforts, it is still not known today whether a deterministic polynomial-time algorithm exists as well. In this paper, we make substantial progress by solving the problem for a multitude of different classes of dense graphs. We solve the Exact Matching problem in deterministic polynomial time for complete r-partite graphs, for unit interval graphs, for bipartite unit interval graphs, for graphs of bounded neighborhood diversity, for chain graphs, and for graphs without a complete bipartite t-hole. We solve the problem in quasi-polynomial time for Erdős-Rényi random graphs G(n, 1/2). We also reprove an earlier result for bounded independence number/bipartite independence number. We use two main tools to obtain these results: A local search algorithm as well as a generalization of an earlier result by Karzanov.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Exact Matching
  • Perfect Matching
  • Red-Blue Matching
  • Bounded Color Matching
  • Local Search
  • Derandomization

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